ab-angle->ABCF A

Average Accuracy: 68.2% → 67.6%

Time: 21.3s

Precision: binary64

Cost: 20681

Math

\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -10000 \lor \neg \left(\frac{angle}{180} \leq 10^{-20}\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
  (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))

↓

(FPCore (a b angle)
 :precision binary64
 (if (or (<= (/ angle 180.0) -10000.0) (not (<= (/ angle 180.0) 1e-20)))
   (+
    (pow b 2.0)
    (* (/ a (/ 2.0 a)) (- 1.0 (cos (* angle (* PI 0.011111111111111112))))))
   (+
    (pow b 2.0)
    (* 3.08641975308642e-5 (* (* a angle) (* PI (* PI (* a angle))))))))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0);
}

↓

double code(double a, double b, double angle) {
	double tmp;
	if (((angle / 180.0) <= -10000.0) || !((angle / 180.0) <= 1e-20)) {
		tmp = pow(b, 2.0) + ((a / (2.0 / a)) * (1.0 - cos((angle * (((double) M_PI) * 0.011111111111111112)))));
	} else {
		tmp = pow(b, 2.0) + (3.08641975308642e-5 * ((a * angle) * (((double) M_PI) * (((double) M_PI) * (a * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0);
}

↓

public static double code(double a, double b, double angle) {
	double tmp;
	if (((angle / 180.0) <= -10000.0) || !((angle / 180.0) <= 1e-20)) {
		tmp = Math.pow(b, 2.0) + ((a / (2.0 / a)) * (1.0 - Math.cos((angle * (Math.PI * 0.011111111111111112)))));
	} else {
		tmp = Math.pow(b, 2.0) + (3.08641975308642e-5 * ((a * angle) * (Math.PI * (Math.PI * (a * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)

↓

def code(a, b, angle):
	tmp = 0
	if ((angle / 180.0) <= -10000.0) or not ((angle / 180.0) <= 1e-20):
		tmp = math.pow(b, 2.0) + ((a / (2.0 / a)) * (1.0 - math.cos((angle * (math.pi * 0.011111111111111112)))))
	else:
		tmp = math.pow(b, 2.0) + (3.08641975308642e-5 * ((a * angle) * (math.pi * (math.pi * (a * angle)))))
	return tmp

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0))
end

↓

function code(a, b, angle)
	tmp = 0.0
	if ((Float64(angle / 180.0) <= -10000.0) || !(Float64(angle / 180.0) <= 1e-20))
		tmp = Float64((b ^ 2.0) + Float64(Float64(a / Float64(2.0 / a)) * Float64(1.0 - cos(Float64(angle * Float64(pi * 0.011111111111111112))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(3.08641975308642e-5 * Float64(Float64(a * angle) * Float64(pi * Float64(pi * Float64(a * angle))))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0);
end

↓

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (((angle / 180.0) <= -10000.0) || ~(((angle / 180.0) <= 1e-20)))
		tmp = (b ^ 2.0) + ((a / (2.0 / a)) * (1.0 - cos((angle * (pi * 0.011111111111111112)))));
	else
		tmp = (b ^ 2.0) + (3.08641975308642e-5 * ((a * angle) * (pi * (pi * (a * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_] := If[Or[LessEqual[N[(angle / 180.0), $MachinePrecision], -10000.0], N[Not[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-20]], $MachinePrecision]], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a / N[(2.0 / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(a * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -1e4 or 9.99999999999999945e-21 < (/.f64 angle 180)

   1. Initial program 30.2%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Simplified 30.2%

      \[\leadsto \color{blue}{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}} \]
      Proof

      [Start] 30.2	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 30.1	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*r/ [<=] 30.2	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 30.2	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
      associate-*r/ [<=] 30.2	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

   3. Taylor expanded in angle around 0 30.6%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   4. Applied egg-rr 29.7%

      \[\leadsto \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 30.6	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 30.6	\[ \color{blue}{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      swap-sqr [=>] 30.5	\[ \color{blue}{\left(a \cdot a\right) \cdot \left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      sin-mult [=>] 29.8	\[ \left(a \cdot a\right) \cdot \color{blue}{\frac{\cos \left(angle \cdot \frac{\pi}{180} - angle \cdot \frac{\pi}{180}\right) - \cos \left(angle \cdot \frac{\pi}{180} + angle \cdot \frac{\pi}{180}\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r/ [=>] 29.8	\[ \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{180} - angle \cdot \frac{\pi}{180}\right) - \cos \left(angle \cdot \frac{\pi}{180} + angle \cdot \frac{\pi}{180}\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]

   5. Simplified 29.8%

      \[\leadsto \color{blue}{\frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 29.7	\[ \frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [<=] 29.7	\[ \frac{\color{blue}{{a}^{2}} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l/ [<=] 29.7	\[ \color{blue}{\frac{{a}^{2}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 29.7	\[ \frac{\color{blue}{a \cdot a}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-/l* [=>] 29.7	\[ \color{blue}{\frac{a}{\frac{2}{a}}} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      mul0-rgt [=>] 29.7	\[ \frac{a}{\frac{2}{a}} \cdot \left(\cos \color{blue}{0} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      cos-0 [=>] 29.7	\[ \frac{a}{\frac{2}{a}} \cdot \left(\color{blue}{1} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 29.8	\[ \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

   if -1e4 < (/.f64 angle 180) < 9.99999999999999945e-21

   1. Initial program 99.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Simplified 99.6%

      \[\leadsto \color{blue}{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}} \]
      Proof

      [Start] 99.6	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 99.6	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*r/ [<=] 99.6	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 99.6	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
      associate-*r/ [<=] 99.6	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

   3. Taylor expanded in angle around 0 99.3%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   4. Taylor expanded in angle around 0 76.5%

      \[\leadsto \color{blue}{\left(-3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, angle \cdot \left(angle \cdot \left({\pi}^{2} \cdot \left(a \cdot a\right)\right)\right), -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({\pi}^{4} \cdot \left(a \cdot a\right)\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 76.5	\[ \left(-3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      +-commutative [=>] 76.5	\[ \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) + -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      fma-def [=>] 76.5	\[ \color{blue}{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right), -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 76.5	\[ \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{\left(angle \cdot angle\right)} \cdot \left({a}^{2} \cdot {\pi}^{2}\right), -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 79.1	\[ \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{angle \cdot \left(angle \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)}, -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 79.1	\[ \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, angle \cdot \left(angle \cdot \color{blue}{\left({\pi}^{2} \cdot {a}^{2}\right)}\right), -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 79.1	\[ \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, angle \cdot \left(angle \cdot \left({\pi}^{2} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right), -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({a}^{2} \cdot {\pi}^{4}\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 79.1	\[ \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, angle \cdot \left(angle \cdot \left({\pi}^{2} \cdot \left(a \cdot a\right)\right)\right), -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \color{blue}{\left({\pi}^{4} \cdot {a}^{2}\right)}\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 79.1	\[ \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, angle \cdot \left(angle \cdot \left({\pi}^{2} \cdot \left(a \cdot a\right)\right)\right), -3.175328964080679 \cdot 10^{-10} \cdot \left({angle}^{4} \cdot \left({\pi}^{4} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]

   6. Taylor expanded in angle around 0 76.5%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

   7. Simplified 98.8%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 76.5	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 76.5	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \color{blue}{\left({\pi}^{2} \cdot {a}^{2}\right)}\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 76.6	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 76.6	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 76.6	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]
      unswap-sqr [=>] 76.6	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 76.6	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]
      swap-sqr [<=] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [<=] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [<=] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [<=] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

   8. Applied egg-rr 98.9%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right) \cdot \left(a \cdot angle\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 98.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 98.9	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)} \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 98.9	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 98.9	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 98.9	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}\right) \cdot \left(a \cdot angle\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 98.9	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}\right) \cdot \left(a \cdot angle\right)\right) + {\left(b \cdot 1\right)}^{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -10000 \lor \neg \left(\frac{angle}{180} \leq 10^{-20}\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	68.2%
Cost	26240

\[{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {b}^{2} \]

Alternative 2
Accuracy	68.1%
Cost	26240

\[{b}^{2} + {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} \]

Alternative 3
Accuracy	59.4%
Cost	20096

\[{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\right) \]

Alternative 4
Accuracy	59.4%
Cost	19840

\[{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2} \]

Alternative 5
Accuracy	59.5%
Cost	19840

\[{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2023138 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))